A compressor \(\mathsf {Q}\)  is unbiased if \(\mathbb {E}[\mathsf {Q}(\mathbf {x})]=\mathbf {x}\)  for all \(\mathbf {x}\in \mathbb {R}^d\) . Though unbiasedness is essential to the linear convergence in [1]}, [2]}, [3]}, the widely-used deterministic quantizers are biased. Round-off errors are inevitable in a finite-precision processor and result in a biased compressor, which satisfies both assumptions when \(\mathbf {x}\)  is large, but only satisfies assum3 if \(\mathbf {x}\)  lies in the subnormal range (e.g. \(|\mathbf {x}|<2^{-126}\)  for single-precision, IEEE 754 standard [4]}). In this view, biased compressors have more practical significance. Some commonly used compressors are given below.
